Which official language of Pakistan is similar to Hindi and written in Persian script?

No native speaker of the Urdu language—myself included—ever calls it Hindustani. The British, in the constitution of the erstwhile British India, called it Urdu as well, never Hindustani. Modern standard Hindi, however, could be described as a highly Sanskritized, highly 'de-Arabicized/de-Persianized' version of Urdu. It's not just the script and the vocabulary, but also pronunciation, idiom, sometimes grammar, and the overall auditory feel of listening to pure Urdu that's notably different from that of Hindi.